Homework Set 1

Problem 1: Expectation values of spin in pure vs. mixed quantum states

A researcher in spintronics is investigated two devices in order to generate spin-polarized currents. One of those devices has spins comprising the current described by the density matrix:

${\hat {\rho }}_{1}={\frac {|\!\!\uparrow \rangle \langle \uparrow \!\!|+|\!\!\downarrow \rangle \langle \downarrow \!\!|}{2}}$,

while the spins comprising the current in the other device are described by the density matrix

${\hat {\rho }}_{2}=|u\rangle \langle u|$ , where $\ |u\rangle ={\frac {e^{i\alpha }|\!\!\uparrow \rangle +e^{i\beta }|\!\!\downarrow \rangle }{\sqrt {2}}}$.

Here $|\!\!\uparrow \rangle$ and $|\!\!\downarrow \rangle$ are the eigenstates of the Pauli spin matrix ${\hat {\sigma }}_{z}$:

${\hat {\sigma }}_{z}|\!\!\uparrow \rangle =+1|\!\!\uparrow \rangle ,\ {\hat {\sigma }}_{z}|\!\!\downarrow \rangle =-1|\!\!\downarrow \rangle$.

What is the spin-polarization of these two currents? Comment on the physical meaning of the difference between the spin state transported by two currents.

HINT: Compute the x, y, and z components of the spin polarization vector using both of these density matrices following the quantum-mechanical definition of expectation value Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle P_{x,y,z} = \langle \hat{\sigma}_{x,y,z}\rangle =\mathrm{Tr}\, [\hat{\rho} \hat{\sigma}_{x,y,z}] } . The colloquial "spin-polarization" discussed in spintronics literature is Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle |\mathbf{P}| } in this rigorous description.

Problem 2: Dynamics of the Bloch vector

The Hamiltonian of a single spin of an electron in external magnetic field Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{B} } is given by (assuming that gyromagnetic ration is unity):

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{H} = -\frac{\hbar}{2} \mathbf{B} \cdot \hat{\boldsymbol{\sigma}} }

where ${\boldsymbol {\sigma }}=({\hat {\sigma }}_{x},{\hat {\sigma }}_{y},{\hat {\sigma }}_{z})$ is the vector of the Pauli matrices. Show that the von Neumann equation of motion

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{\partial \hat{\rho}}{\partial t} = -\frac{i}{\hbar} [\hat{H},\hat{\rho}] }

for the density matrix of spin-1/2

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{\rho} = \frac{1}{2} \left( \hat{I} + \mathbf{P} \cdot \hat{\boldsymbol{\sigma}} \right) }

can be recast into the equation of motion for the Bloch (or spin-polarization) vector because Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{\mathbf{\rho}} } and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{P} } are in one-to-one correspondence in the case of spin-1/2. Find explicitly the right hand side of such an equation:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \frac{d \mathbf{P}}{dt} = ? }

HINT: Use the following property of the Pauli matrices:

${\hat {\sigma }}_{\alpha }{\hat {\sigma }}_{\beta }-{\hat {\sigma }}_{\beta }{\hat {\sigma }}_{\alpha }=2i\epsilon _{\alpha \beta \gamma }{\hat {\sigma }}_{\gamma }$.

Problem 3: Does entropy increase in a closed quantum system?

In classical Hamiltonian systems the nonequilibrium entropy

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle S = -k_B \int \rho \ln \rho }

is constant in time. In this problem we want to demonstrate that in microscopic evolution of an isolated quantum system, the entropy is also time independent, even for general time-dependent density matrix Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{\rho} } . That is, using the equation of motion:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle i \hbar \frac{\partial \hat{\rho}}{\partial t} = [\hat{H},\hat{\rho}] }

prove that von Neumann entropy

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle S(t) =-k_B \mathrm{Tr}[\hat{\rho}(t) \ln \hat{\rho}(t)] }

is time independent for arbitrary density matrix ${\hat {\rho }}(t)$.

HINT: Use Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{Tr}(\hat{A}\hat{B}\hat{C})=\mathrm{Tr}(\hat{C}\hat{A}\hat{B}) } for any operators Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{A} } , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{B} } , Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{C} } , as well as that an operator ${\hat {M}}$ commutes with any function Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle f(\hat{M}) } :

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle [\hat{M},f(\hat{M})]=0 } .

Problem 4: Successive measurements on subsystems of composite bipartite quantum system

Consider a quantum system composed of two spins, labeled as subsystem A and B. The quantum state of the composite system is described by the following density matrix in the Hilbert space Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathcal{H}_A \otimes \mathcal{H}_B } :

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{\rho} = \frac{1}{8} \hat{I} + \frac{1}{2} |\Psi\rangle \langle \Psi| }

where ${\hat {I}}$ denotes the Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 4 \times 4 } unit matrix in Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathcal{H}_A \otimes \mathcal{H}_B } and

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle |\Psi \rangle = \frac{1}{\sqrt{2}}\left( |\!\!\uparrow \rangle \otimes |\!\!\downarrow \rangle - |\!\!\downarrow \rangle \otimes |\!\!\uparrow \rangle \right) }

is entangled state (in the context of spins also called "singlet") of two spins.

Suppose we measure the first spin (subsystem A) along the axis described by the unit vector Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{n} } , and the second spin (subsystem B) along the axis described by the unit vector $\mathbf {m}$, where Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{n} \cdot \mathbf{m} = \cos \theta } . What is the probability that both spins are "spin-up" along their respective axes?

HINT: In general, the probability to measure eigenvalue Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lambda } of a physical quantity in the quantum state described by the density matrix Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{\rho} } is given by $\mathrm {prob} =\mathrm {Tr} [{\hat {\rho }}{\hat {P}}_{\lambda }]$. Here Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{P}_\lambda } is the projection operator on the eigensubspace corresponding to eigenvalue Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \lambda } . To find the probability of measurement on the subsystem, one should use the density matrix of that subsystem, obtained by partial trace over the states of the second subsystem. This means that the probability asked in the problem is defined by:

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{prob} = \mathrm{Tr}_B \left\{ \hat{P}_\mathbf{m}^B \hat{\rho}^B \right\} = \mathrm{Tr}_B \left\{ \hat{P}_\mathbf{m}^B \mathrm{Tr}_A \left[(\hat{P}_\mathbf{n}^A \otimes \hat{I}^B) \hat{\rho} \right] \right\} } .

The eigenprojector for the "spin-up" (i.e., +1) eigenvalue along the Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{n} } -axis is simply:

${\hat {P}}_{\mathbf {n} }^{A}=|\!\!\uparrow _{\mathbf {n} }\rangle \langle \uparrow _{\mathbf {n} }\!\!|={\frac {1}{2}}\left({\hat {I}}^{A}+\mathbf {n} \cdot {\hat {\boldsymbol {\sigma }}}^{A}\right)$.

We also use the fact that resulting state of the composite system after the selective measurement on subsystem A is described by the density matrix ${\hat {P}}_{\mathbf {n} }^{A}{\hat {\rho }}{\hat {P}}_{\mathbf {n} }^{A}$, so that subsystem B after the measurement on system A is "collapsed" onto the state described by the density matrix Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{\rho}^B = \mathrm{Tr}_A [\hat{P}_\mathbf{n}^A \hat{\rho} \hat{P}_\mathbf{n}^A]= \mathrm{Tr}_A [\hat{P}_\mathbf{n}^A \hat{\rho}] } , where we use cyclic property of the trace and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (\hat{P}_\mathbf{n}^A)^2=\hat{P}_\mathbf{n}^A } .

Problem 5: Ambiguity of ensemble decomposition of density operator

(a) Suppose we have a statistical mixture of spin-1/2 particles that consists of the state Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle |\uparrow_z \rangle } with probability $(1+1/{\sqrt {2}})/2$ and the state Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle |\downarrow_z \rangle } with probability Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle (1 - 1/\sqrt{2})/2 } . Find the matrix representation of the density operator (i.e., the density matrix) ${\hat {\rho }}$ in the basis of states Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \{|\uparrow_z \rangle , |\downarrow_z \rangle \} } , as well as in the basis of eigenstates of the Pauli matrix ${\hat {\sigma }}_{x}$ denoted as $\{|\uparrow _{x}\rangle ,|\downarrow _{x}\rangle \}$. Compute purity Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathrm{Tr}\, \hat{\rho}^2 } of this density matrix and its Bloch vector, and comment if this quantum state is pure or mixed.

(b) Now suppose that we have a mixed state with Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 1/2 } probability to have spin pointing along Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{e}_{n_1} = (\mathbf{e}_z + \mathbf{e}_x)/\sqrt{2} } and Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle 1/2 } probability to have spin along Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \mathbf{e}_{n_1} = (\mathbf{e}_z - \mathbf{e}_x)/\sqrt{2} } .

(c) Compare state in (a) to the one in (b), including comparing their Bloch vectors. Are they the same or different?

HISTORICAL NOTE: The general theorem on when two different mixtures of pure quantum states represent the same density matrix Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{\rho} } :

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \hat{\rho} = \sum_i p_i |\Psi_i\rangle \langle \Psi_i| = \sum_j q_j |\Phi_i\rangle \langle \Phi_i| }

such as that

Failed to parse (SVG with PNG fallback (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://en.wikipedia.org/api/rest_v1/":): {\displaystyle \sqrt{q_j} |\Phi_j \rangle = \sum_i U_{ji} \sqrt{p_i} |\Psi_i \rangle }

was proven by Schrödinger in 1936 (he commented "this theorem was one for which I claim no priority but the permission of deducing it in the following section, for it is certainly not well known"), and then rediscovered by Hughston, Jozsa, and Wootters (HJW) in 1993. In 1998, Mermin simplified a portion of HJW’s proof, commenting that on it as "a pertinent theorem which deserves to be more widely known", while none of them knew about the original Schrödinger work (so, today it is proper to denote it as Schrödinger-HJW theorem).